Perturbative spectra in gauge theories with gravity duals Christoph Sieg Niels Bohr International Academy Niels Bohr Institute 22.02.10, Nordic String Meeting, Hannover C.S., A. Torrielli: 0505071 F. Fiamberti, A. Santambrogio, C.S., D. Zanon: 0712.3522 0806.2095 0806.2103 0811.4594 F. Fiamberti, A. Santambrogio, C.S.: 0908.0234 J. Minahan, O. Ohlsson Sax, C.S.: 0908.2463 0912.3460
Outline Introduction and overview Perturbative calculations Conclusions and outlook
AdS / CFT correspondence energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O L ∝ tr φ i 1 . . . φ i L integrable systems [Bena, Polchinski, Roiban] [Kazakov, Marshakov, [Minahan, Zarembo] (asymptot.) Bethe ansätze Minahan, Zarembo] [Beisert, Dippel, Staudacher] string gauge [Arutyunov, Frolov, Staudacher] dressing phase [Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher]
AdS / CFT correspondence energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O S ∝ tr φ D S φ integrable systems [Bena, Polchinski, Roiban] [Kazakov, Marshakov, [Minahan, Zarembo] (asymptot.) Bethe ansätze Minahan, Zarembo] [Beisert, Dippel, Staudacher] string gauge [Arutyunov, Frolov, Staudacher] [ D , Q i ] = 0 dressing phase [Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher] dilatation operator [Beisert, Kristjansen, Staudacher] D � O L = γ � O L Feynman graph computations in the flavour SU ( 2 ) subsector: 1-loop: [Berenstein, Maldacena, Nastase] 2-loops: [Gross, Mikhailov, Roiban] checks at higher loops: [Gross, Mikhailov, Roiban] [Beisert, McLoughlin, Roiban] [Fiamberti, Santambrogio, CS, Zanon] [Fiamberti, Santambrogio, CS]
AdS / CFT correspondence energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O S ∝ tr φ D S φ integrable systems [Bena, Polchinski, Roiban] [Kazakov, Marshakov, [Minahan, Zarembo] (asymptot.) Bethe ansätze Minahan, Zarembo] [Beisert, Dippel, Staudacher] string gauge [Arutyunov, Frolov, Staudacher] dressing phase [Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher] [Gubser, Klebanov, Polyakov] [Kotikov, Lipatov, Velizhanin [Frolov, Tseytlin] [Kruczenski] [Makeenko] S ≫ 1 integral eq. for f ( λ ) √ 2 π 2 − ζ ( 2 ) λ 2 π − 3 λ λ π ln 2 λ ≫ 1 λ ≪ 1 16 π 4 [Eden, Staudacher] [Benna, Benvenuti, Klebanov, Sardicchio] [Casteill, Kristjansen] [Alday, Arutyunov, Benna, Eden, Klebanov] [Basso, Korchemsky, Kotanski]
AdS / CFT correspondence energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O S ∝ tr φ D S φ integrable systems [Bena, Polchinski, Roiban] [Kazakov, Marshakov, [Minahan, Zarembo] (asymptot.) Bethe ansätze Minahan, Zarembo] [Beisert, Dippel, Staudacher] string gauge [Arutyunov, Frolov, Staudacher] dressing phase [Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher] S = 2 Konishi multiplet tr φ i φ i tr φ D 2 φ tr φ [ φ, Z ] Z � �� � L = 4 � γ asy = λ k γ k k < L
AdS / CFT correspondence energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O S ∝ tr φ D S φ integrable systems [Bena, Polchinski, Roiban] [Kazakov, Marshakov, [Minahan, Zarembo] (asymptot.) Bethe ansätze Minahan, Zarembo] [Beisert, Dippel, Staudacher] string gauge [Arutyunov, Frolov, Staudacher] dressing phase [Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher] S = 2 wrapping interactions finite size effects [C.S.,Torrielli] [Ambjørn, Janik, Kristjansen] [Schafer-Nameki] Konishi multiplet [Janik, Lukowski] tr φ i φ i tr φ D 2 φ tr φ [ φ, Z ] Z � �� � thermodyn. Bethe ansatz L = 4 L = 4 loop [Arutyunov, Frolov] � Y-system γ asy = λ k γ k Feynman graphs [Gromov, Kazakov, Vieira] [Fiamberti,Santambrogio,C.S.,Zanon] k < L (gen.) Lüscher formula [Bajnok,Janik] γ = γ asy + λ 4 γ 4 + . . .
AdS 4 / CFT 3 (ABJM) correspondence [Aharony, Bergman, Jafferis, Maldacena] Type II A ST AdS 4 × CP 3 ⇔ 3-dim. N = 6 CS theory energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O J ∝ tr φ n 1 . . . φ n J integrable systems [Arutyunov, Frolov] [Minahan, Zarembo] [Stefanski] (asymptot.) Bethe ansätze [Bak, Rey] [Gromov, Vieira] [Gromov, Vieira] [Ahn, Nepomechie] magnon dispersion relation [Beisert, Dippel, Staudacher] [Beisert] [Arutyunov, Frolov, Zamaklar] � Q 2 + 4 h ( λ ) 2 sin 2 p E = 2 − Q h ( λ ) 2 = ? h ( λ ) 2 = λ 4 π 2 unexpectedly simple in AdS 5 / CFT 4
AdS 4 / CFT 3 (ABJM) correspondence [Aharony, Bergman, Jafferis, Maldacena] Type II A ST AdS 4 × CP 3 ⇔ 3-dim. N = 6 CS theory energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O J ∝ tr φ n 1 . . . φ n J integrable systems [Arutyunov, Frolov] [Minahan, Zarembo] [Stefanski] (asymptot.) Bethe ansätze [Bak, Rey] [Gromov, Vieira] [Gromov, Vieira] [Ahn, Nepomechie] magnon dispersion relation BMN limit [Beisert, Dippel, Staudacher] [Beisert] giant magnons [Arutyunov, Frolov, Zamaklar] � [Nishioka, Takayanagi] Q 2 + 4 h ( λ ) 2 sin 2 p E = 2 − Q [Gaiotto, Giombi, Yin] [Grignani, Harmark, Orselli] two loops quantum corr. [Nishioka, Takayanagi] [McLoughlin, Roiban, Tseytlin] [Minahan, Zarembo] [Bak, Rey] � h ( λ ) 2 = ? λ λ ln 2 λ 2 2 − 2 π h ( λ ) 2 = λ 4 π 2 unexpectedly simple in AdS 5 / CFT 4
AdS 4 / CFT 3 (ABJM) correspondence [Aharony, Bergman, Jafferis, Maldacena] Type II A ST AdS 4 × CP 3 ⇔ 3-dim. N = 6 CS theory energy E anom. dim. γ λ ≫ 1 λ ≪ 1 (semicl.) strings comp. operators O J ∝ tr φ n 1 . . . φ n J integrable systems [Arutyunov, Frolov] [Minahan, Zarembo] [Stefanski] (asymptot.) Bethe ansätze [Bak, Rey] [Gromov, Vieira] [Gromov, Vieira] [Ahn, Nepomechie] magnon dispersion relation BMN limit [Beisert, Dippel, Staudacher] four loops [Beisert] giant magnons [Arutyunov, Frolov, Zamaklar] [Minahan, Ohlsson Sax, C.S.] � [Nishioka, Takayanagi] Q 2 + 4 h ( λ ) 2 sin 2 p E = 2 − Q [Gaiotto, Giombi, Yin] [Grignani, Harmark, Orselli] two loops quantum corr. [Nishioka, Takayanagi] [McLoughlin, Roiban, Tseytlin] [Minahan, Zarembo] [Bak, Rey] � h ( λ ) 2 = ? λ 2 + λ 4 ( − 16 + 4 ζ ( 2 )) λ λ ln 2 2 − 2 π h ( λ ) 2 = λ 4 π 2 unexpectedly simple in AdS 5 / CFT 4
Renormalization of composite operators � composite operator O L = L of length L (with L scalar fields) two-point functions of composite operators: tree level δ AB O A L ( x ) , 1 , O B � L ( y ) � ∆ = L = = ( x − y ) 2 ∆ , x y
Renormalization of composite operators � composite operator O L = L of length L (with L scalar fields) two-point functions of composite operators: with loop corrections δ AB O A L ( x ) , V 2 L , O B � L ( y ) � ∆ = L + γ + . . . = V = ( x − y ) 2 ∆ , x y
Renormalization of composite operators � composite operator O L = L of length L (with L scalar fields) two-point functions of composite operators: with loop corrections δ AB O A L ( x ) , V 2 L , O B � L ( y ) � ∆ = L + γ + . . . = V = ( x − y ) 2 ∆ , x y renormalization of composite operators in a CFT in D = 4 − 2 ε dimensions D = µ d O a L , ren = Z a b O b d µ ln Z ( λµ 2 ε ) L , bare ,
Renormalization of composite operators � composite operator O L = L of length L (with L scalar fields) two-point functions of composite operators: with loop corrections δ AB O A L ( x ) , V 2 L , O B � L ( y ) � ∆ = L + γ + . . . = V = ( x − y ) 2 ∆ , x y renormalization of composite operators in a CFT in D = 4 − 2 ε dimensions D = µ d O a L , ren = Z a b O b d µ ln Z ( λµ 2 ε ) L , bare , anomalous dimensions: � λ k D k eigenvalues of the dilatation operator D = k ≥ 1 D � O L = γ � O L
Bethe ansatz in the flavour SU ( 2 ) subsector 2 ( φ 1 + i φ 2 ) , ψ = 2 ( φ 3 + i φ 4 ) , Z = 2 ( φ 5 + i φ 6 ) 1 1 1 complex fields: φ = √ √ √ ψ only as internal flavour in Feynman diagrams
Bethe ansatz in the flavour SU ( 2 ) subsector 2 ( φ 1 + i φ 2 ) , ψ = 2 ( φ 3 + i φ 4 ) , Z = 2 ( φ 5 + i φ 6 ) 1 1 1 complex fields: φ = √ √ √ ψ only as internal flavour in Feynman diagrams map to integrable spin chain of length L ZZZ . . . Z O L = tr ( φ . . . φ ) ↔ � �� � � �� � M L − M BPS operator tr ( Z . . . Z ) ↔ ferromagnetic vaccum impurities φ ↔ spin excitations (magnons) Hamiltonian H dilatation operator D ↔ energies E ↔ anomalous dimensions γ
Bethe ansatz in the flavour SU ( 2 ) subsector 2 ( φ 1 + i φ 2 ) , ψ = 2 ( φ 3 + i φ 4 ) , Z = 2 ( φ 5 + i φ 6 ) 1 1 1 complex fields: φ = √ √ √ ψ only as internal flavour in Feynman diagrams map to integrable spin chain of length L ZZZ . . . Z O L = tr ( φ . . . φ ) ↔ � �� � � �� � M L − M BPS operator tr ( Z . . . Z ) ↔ ferromagnetic vaccum impurities φ ↔ spin excitations (magnons) Hamiltonian H = H XXX 1 / 2 + . . . dilatation operator D ↔ energies E ↔ anomalous dimensions γ operator mixing problem solved by the asymptotic Bethe ansatz M M M S ( u j , u k ) e 2 i θ ( u j , u k ) , E = �� � π 2 sin 2 p j � � � p j = 0 , e ip j L = ˆ 1 + λ 2 − 1 j = 1 k � = j j = 1 matrix part dressing phase momentum single magnon conservation dispersion relation two-particle S-matrix
b b b b b One loop � � � � i tr ( ψ [ Z ,φ ]) = i − i tr ( ¯ ψ [¯ φ, ¯ Z ]) = − i − , − = − = =
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